A remedy for major cosmological tensions: Dark energy with an oscillating inertial mass density

Abstract

The preponderance of observational evidence indicates that a vast portion of the energy density of the Universe today comes in dark matter and enigmatic dark energy (DE). The standard cosmological model, namely, the so-called Lambda Cold Dark Matter model ($\Lambda$CDM), resting on this dark sector as well as a small fraction of baryons has been remarkably successful in elucidating the bulk of Universe we inhabit. Though, astronomical observations improving in precision over the course of years are increasingly exposing that $\Lambda$CDM is significantly discrepant with various datasets. The direct and local measurements of the present-day expansion rate yielding $H_{0}=73.04\pm1.04 \;\, {\rm km\,s^{-1}\,Mpc}^{-1}$ are at more than $5\sigma$ tension (the Hubble $H_0$ tension) with the one $H_{0}=67.36\pm0.54 \;\, {\rm km\,s^{-1}\,Mpc}^{-1}$ inferred within the $\Lambda$CDM based on matter-baryon densities and the spacing between acoustic peaks of the CMB. The $H_0$ tension effectively propagates to the supernovae absolute magnitude $M_B$ through the distance modulus $\mu(z_i,H(z))=m_{B,i}-M_{B,i}$ where $m_{B,i}$ is the measured apparent magnitude of the supernovae observed at the redshift $z_i$, and creates a $3.4\sigma$ tension with the results calibrated by the CMB sound horizon scale. Another discrepancy regarding the expansion rate $H(z)$ within the best fit $\Lambda$CDM is the $\sim1.5\sigma$ tension between low (Galaxy BAO) and high redshift (Lyman-$\alpha$ at $z\approx2.33$) BAO data. It first emerged as a preference for smaller $H(z)$ and accompanying negative DE densities for $1.7\lesssim z\lesssim2.34$, being at $2.5\sigma$ tension with $\Lambda$CDM. In addition, $\Lambda$CDM predicts a larger weighted amplitude of matter fluctuations $S_8$ in comparison with what the independent large scale structure dynamical low-redshift probes suggest, thereby running into $2$ to $3\sigma$ tension. Given the long-standing theoretical issues such as the cosmological constant and coincidence problems related to the $\Lambda$, all these enumerated challenges and more inevitably motivate many to seek for a more complete framework either as modified theories of gravity or as minimal extensions beyond $\Lambda$CDM in the context of General Relativity (GR). In this sense, an approach that constitutes an example of the latter attempts focuses on inertial mass density $\Varrho\;=\rho+p$ parametrizations. The graduated dark energy (gDE) model proposed in Akarsu \textit{et al}. [Phys. Rev. D 101, 063528 (2020)], whose inertial mass density $\Varrho$ measures the minimum dynamical deviation $\Varrho\;\propto \rho^{\lambda}$ from the assumption of null QFT vacuum energy $\Varrho_{\Lambda}\;=0$ is one that exhibits nontrivial properties. It turns out that smaller and smaller negative values of the parameter $\lambda$ corresponds to a constant negative DE density that changes its sign from negative to positive in the past. Such a dynamical behavior would imply that $H(z)$ suppressed by the presence of a negative source at high redshifts results in an enhanced $H(z)$ at lower redshifts as the comoving angular diameter distance $D_M(z)$ to the surface of last scattering $D_M(z_*)=c\int_0^{z_*} H^{-1}(z)\d z$ is very stringently and almost model-independently constrained by the CMB for any given pre-recombination physics and should be kept unaltered. This means that if the redshift at which the sign-flip in the DE density occurs is slightly below the anomalous Ly-$\alpha$ at effective redshift $z\approx2.34$, it is quite conceivable that a dynamical DE possessing negative energy density mitigates both the $H_0$ and Ly-$\alpha$ discrepancies. The observational analysis of the gDE strongly favors a scenario in which the sign change of the gDE density is so swift it is practically identical to the cosmological constant phenomenologically flipping its sign much like the step function, except that $\Lambda<0$ in the past. It arises as a limiting case $\lambda\rightarrow-\infty$ of the gDE such that $\rho_{\rm gDE}(a)\rightarrow \rho_{\rm gDE,0}{\rm sgn}[f(a)]$ where sgn is the signum function. This limit has been comprehensively studied under the name of the $\Lambda_{\rm s}$CDM model in Akarsu \textit{et al}. [Phys. Rev. D 104, 123512 (2021)] where the late-time accelerated expansion is driven by $\Lambda_s\equiv\Lambda_{\rm s0}{\rm sgn}[z-z_{\dagger}]$ with $z_\dagger$ being the switching redshift rather than the usual $\Lambda$. The confrontation with the data sets encompassing CMB, Pantheon SNIa with and without SH0ES $M_B$ priors and BAO, shows that $\Lambda_{\rm s}$CDM simultaneously ameliorates six of the present significant tensions, namely $H_0$, $M_B$, $S_8$, Ly-$\alpha$, $t_0$ and $\omega_b$ tensions.